1. The problem statement, all variables and given/known data
is there are a set with countably infinite number of accumulation points?

2. Relevant equations

is there a set with exactly two distinct accumulation points?

3. The attempt at a solution
a set with two accumulation points might be: {1/n + (-1)^n}
i have no clue about the countably infinite one

Your idea is fine, but be sure to write it correctly (with an ": n in N").

I'd prefer not to spoon feed you an example for the other; rather, consider perhaps a set (in R) that depends on two natural numbers. Say, [tex]\{y_n, y_n + x_m: n, m \in \mathbb{N}\}[/tex], which you might construct so that [tex]\lim_{m\to\infty} x_m = 0[/tex] and so, for each fixed n, [tex]\lim_{m\to\infty} y_n+x_m=y_n.[/tex]